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Localized fields on scalar global defects F.A. Britoa and H. Henriquea,b aDepartamento de F´ısica, Universidade Federal de Campina Grande, Caixa Postal 10071, 58109-970 Campina Grande, Para´ıba, Brazil bDepartamento de F´ısica Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, Rio Grande do Norte, Brazil Weinvestigatethelocalization ofmodesontheworldvolumeofap-braneembeddedinp+d+1- dimensionalspacetime. Thep-branehereissuchthatitsprofileisregardedasascalarglobaldefect andthemodeslocalized are scalar modes thatcome from thefluctuationsaround such defect. The effectiveactiononthebraneiscomputedandtheinducedpotentialsaretypicallyφ4-typepotentials thatareflatterforlowerd-dimensions. Wealsomakeaconnectionofsuchscalarglobaldefectswith black p-branesin certain limits. 1 1 Inthisworkweinvestigateacompactificationmechanismthroughlocalizationoffieldsmodesonp-braneembedded 0 in p+d+1-dimensional spacetime with topology p+d+1 =Mp+1 Rd. The p-brane here is such that its profile is 2 M × regardedasascalarglobaldefectandthemodeslocalizedarescalarmodesthatcomefromthefluctuationsaroundthe n scalarglobaldefects firstintroducedin [1]. These modes aredescribedvia eigenfunctions thatsatisfy a Schroedinger- a like equation for the fluctuations. We then integrate out these modes in the internal space Rd to obtain an effective J action describing the fields localized on the scalar global defect (the p-brane) following the lines of the references 6 [2–5]. Such defects are scalar soliton solutions of a scalar field theory in arbitrary dimensions even for d > 2. This is known to evade the Derrick’s theorem with the price of being a theory that breaks translational invariance. They ] h are different of the co-dimensional one objects mostly used in the braneworlds scenarios [6–17]. The scalar potential -t in the Lagrangian now depends explicitly on the spatial coordinates. However, we show that the effective theory p describing localized scalar modes on the global defect worldvolume recovers the translational invariance because the e effective potentials has no dependence on any spatial coordinate on the p-brane. In this sense we conclude that h in this compactification process arises a mechanism in which a theory with broken translation invariance in higher [ dimensions recoversthe translationalinvariance in lowerdimensions. This is in accordwith the recentconsiderations 1 on spacetime symmetries broken in high energy probing extra dimensional physics in our Universe. v Consider a theory of a scalar field embedded in a p+d+1-dimensional spacetime with topology 1 7 p+d+1 =Mp+1 Rd, (1) 2 M × 1 whereM =(yµ,q), withq =(x ,x ,...,x )being coordinatesinthe internalflatspaceandyµ =(t,yi) (i=1,2,...,p) 1 2 d . arecoordinatesofthe p-braneworldvolumeembeddedina (p+d+1)- dimensionalspacetime. The actionis givenby 1 0 1 1 S = dp+1yddq ∂ φ∂Mφ V(φ) , (2) M 1 2 − Z (cid:20) (cid:21) : v that can be written in a more convenient form as follows i X 1 ∂φ 2 p ∂φ 2 S = dp+1yddq ( φ)2 V(φ) , (3) ar Z 2(cid:18)∂t(cid:19) − Xi=1 ∂yi! − ∇q −    with ddq =dx dx ...dx =rd−1drdΩ , being Ω = 2πd/2 the (d 1)-dimensionalvolume, of the unit (d 1)- 1 2 d (d−1) (d−1) Γ(d/2) − − sphere. Let us now apply the perturbation theory to the scalar field φ as φ(q,yµ) φ¯(q)+η(q,yµ), (4) −→ such that S(φ) S(φ¯,η). (5) −→ This allows us to expand the potential around the static solution describing the scalar globaldefect. The action now reads S = dp+1yddq 1 φ¯ 2 V(φ¯) 1∂ η∂µη q µ −2 ∇ − − 2 Z (cid:26) 1 (cid:0) (cid:1)V′′′(φ¯) V′′′′(φ¯) + η 2η ηV′′(φ¯)η η3 η4+... , (6) 2 ∇q − − 3! − 4! (cid:27) (cid:0) (cid:1) 2 thatisanactionforthe fluctuationsη ofthep-brane. Sinceweassumethe scalarfieldandfluctuationswithspherical symmetry, the Laplacian is simply given in terms of the radial coordinate 1 d d 2 = rd−1 . (7) ∇q rd−1dr dr (cid:18) (cid:19) Now, by using this action, we can identify two important terms: the tension T of the p-brane and the bilinear p Hamiltonian operator ηHη, i.e., 1 T = ddq ( φ¯)2+V(φ¯) (8) p q 2 ∇ Z (cid:20) (cid:21) and 1 1 ηHη = η( 2+V′′(φ¯))η. (9) 2 2 −∇q Thus, 1 1 S = dp+1yT + dp+1yddq ∂ η∂µη ηHη p µ − −2 − 2 Z Z (cid:20) V′′′(φ¯) V′′′′(φ¯) η3 η4+... . (10) − 3! − 4! (cid:21) This action allows us to write a Lagrangianfor the fluctuations, given by 1 1 = T δd(q)+ ∂ η∂µη ηHη L(p+d+1) p −2 µ − 2 (cid:20) V′′′(φ¯) V′′′′(φ¯) η3 η4+... . (11) − 3! − 4! (cid:21) Using the Euler-Lagrangeequation ∂ ∂ (p+d+1) (p+d+1) L ∂ L =0 (12) µ ∂η − ∂(∂ η) (cid:20) µ (cid:21) we find the following equation of motion V′′′(φ¯) V′′′′(φ¯) Hη+ η2+ η3+...=∂ ∂µη (cid:3) η. (13) 2! 3! µ ≡ (p+d) In the linear regime we have Hη =(cid:3) η. (14) (p+d) Now writing the fluctuations in terms of a sum of normal modes we find η(yµ,q)= ξ (y)ψ (q), (15) n n n X that substituting into (14) and assuming that the modes ξ (y) are fields describing localized particles on the p-brane n satisfying a Klein-Gordon equation (cid:3) ξ (y)=M2ξ (y), (16) (p+d) n n n we find a Schroedinger-like equation that governs the masses M2 of the particles given by n 2ψ (q)+V′′(φ¯)ψ (q)=M2ψ (q). (17) −∇q n n n n On the other hand, assuming the following orthogonality condition for the wave functions ψ(q) ddqψ (q)ψ (q)=δ , (18) m n m,n Z 3 substituting (15) into the action (10) and finally integrating in q, we find the effective action N S = dp+1y T + ∂ ξ (y)∂µξ (y)+V(ξ) , (19) p µ n n − " # Z n=0 X where the potential for the localized modes is written as 1 V′′′(φ¯) V′′′′(φ¯) V(ξ)= ddq ηHη+ η3+ η4+... . (20) 2 3! 4! Z (cid:20) (cid:21) We nowfocus onscalarglobaldefects by deforming the usualscalarfieldtheory by consideringthe scalarpotential with an explicit dependence on the spatial coordinates [1] 1 V(φ,r)= W2. (21) 2r2d−2 φ The first derivative of the superpotential (W =∂W/∂φ) is given by φ a−1 a+1 Wφ = φ a φ a . (22) − (cid:16) (cid:17) d > 2 is the dimension of the internal space and a = 1,3,5... is a dimensionless parameter of the theory. To find topological solutions that describe scalar global defects where we shall introduce fluctuations around, we shall make use of the Bogomol’nyi formalism which is useful to reduce equations of motion to first order differential equations that are easier to integrate. Thus, for static fields in d dimensions we find dφ 1 = W . (23) dr ±rd−1 φ By substituting (22) into equation above, for the plus sign, we find the solution 1 r2−d φ¯(r)=tanha . (24) a d 2 (cid:20) (cid:18) − (cid:19)(cid:21) Now before address the issue of localized spectrum on the p-brane, we shall first show that in the present system there is only one bound state, a zero mode given by the eigenfunction ψ , followed by a tower of continuum massive 0 modes that we disregard in the present study. It is not difficult to show that the Schroedinger-like equation can be factored in terms of another operator as follows [1] 1 H = Q†Q (25) r2d−2 with d Q=rd−1 W , (26) φφ dr ∓ and d Q† = rd−1 W . (27) φφ − dr ∓ This guarantees that the operator H is quadratic and so no tachyonic modes are allowed. Furthermore since the Schroedinger potential approaches zero as r then the only bound state is the zero mode ψ . We can determine 0 →∞ such a mode by solving the following eigenvalue equation with M =0 0 Hψ =M2ψ , 0 0 0 that is, 1 d rd−1 W ψ =0, rd−1 dr ∓ φφ 0 (cid:20) (cid:21) 4 whose solution is W φφ ψ =cexp dr , (28) 0 ± rd−1 (cid:20) Z (cid:21) where c is a normalizationconstantandin generaljust one solutionwith a particular signal is normalizable. Thus, ± since we have only one bound state ψ , the effective action (19) reads 0 S = dp+1y[T +∂ ξ (y)∂µξ (y)+V(ξ)], (29) p µ 0 0 − Z the tension (8) is now written as 1 2πd/2 2a T = ddq ( φ¯)2+V(φ¯) = p 2 ∇q Γ(d/2)(4a2 1) Z (cid:20) (cid:21) − 2a = Ω , (30) (d−1)(4a2 1) − and the effective potential (20) in now given by V′′′(φ¯) V′′′′(φ¯) V(ξ )= ddq (ξ ψ )3+ (ξ ψ )4+... . (31) 0 0 0 0 0 3! 4! Z (cid:20) (cid:21) Notice that since we have just a zero mode, the mass term in the potential disappeared. The dots mean higher order terms. These terms are not present for a = 1 since in this case V(φ) is at most of the fourth order, so that this potential turns out to be exact. However, the wave function is normalized only for a>1. In spite of this, we use the case a = 1 to show that our setup has a hidden connection with black p-branes as a solution of the type II supergravity. We do not need fluctuations (the wave function) for the moment. For this proposal let us compare our action (2) with the bosonic sector of type II supergravity action (with p+1+d=10) 1 S = dp+1+dx√g e−2Φ(R+4∂ Φ∂MΦ) F 2 . (32) M p+2 − 2| | Z (cid:20) (cid:21) The extremal black Dp-brane solution is given by r 7−p ds2 =H−1/2η dyµdyν +H1/2(dr2+r2dΩ2 ), H (r)=1+ p , eΦ =g H(3−p)/4, (33) p µν p 8−p p r s p (cid:16) (cid:17) with the R-R field strength givenby F =dH−1 dx0 dx1 ... dxp. In the limit r r , H 1, the dilaton Φ p+2 p ∧ ∧ ∧ ∧ ≫ p p → approaches to a constant and F 1/r8−p. Now substituting this solution into the action (32) we find p+2 | |∼ 1 dr S = Ω dp+1y . (34) −2 8−p r8−p Z Z The same action can be found from (2) by considering a=1 such that V =W2/2r2d−2 =(1 φ2)2/2r2d−2 with the φ − solution (24) gives 1 1 r2−d V = sech4 . (35) 2r2d−2 a d 2 (cid:20) (cid:18) − (cid:19)(cid:21) For r we find V 1/2r2d−2 and φ¯ 1/rd−2, such that substituting this solution into (2) we find →∞ → ∼ 1 dr S = Ω dp+1y . (36) −2 d−1 rd−1 Z Z Of course, the actions (34) and (36) are the same for p+d+1 = 10. Notice that the heaviest p-branes, for fixed a, occur for d 7. Being M = T V , the entropy of the corresponding black p-brane S M2 is also maximal around p p ≃ ∼ this dimension. Letusnowstudythecaseswitha>1. Theeffectivepotentialdevelopsξ3-termsonlyfora=2,butinthefollowing we discuss only cases for ξ4-terms. We shall focus only on the cases d=3 and d=6 with a=3 to compare flatness of the induced ξ4-terms in the effective potential which is essential for cosmological issues. The effective action for d=3 and a=3 is found by considering the following results: 5 Static solution • 1 φ¯=tanh3 (37) 3r (cid:18) (cid:19) Normalized wave-function • 1 1 Ψ =2.057tanh2 sech2 (38) 0 3r 3r (cid:18) (cid:19) (cid:18) (cid:19) p-brane tension • 24 T = π (39) p 35 Effective potential • V(ξ )=0.95πξ4. (40) 0 0 Similarly, the effective action for d=6 and a=3 is found by considering the following results: Static solution • 1 φ¯=tanh5 (41) 5r (cid:18) (cid:19) Normalized wave-function • 1 1 Ψ =6.7009tanh2 sech2 (42) 0 12r4 12r4 (cid:18) (cid:19) (cid:18) (cid:19) p-brane tension • 6 T = π3 (43) p 35 Effective potential • V(ξ )=26.72π3ξ4. (44) 0 0 The Fig. 1 shows how flat the effective potential on the p-brane is as a function of the number of dimensions d. In summary we conclude that although the scalar global defects break the translation symmetry because the scalar potential has explicit dependence with the spatial coordinates, it is possible to find effective theories living on the scalar global defect (the p-brane) worldvolume whose effective potentials are invariant under such symmetry. This is so because such potentials have no explicit dependence with the coordinates of the p-brane, thus they may respect the translationsymmetryalongthe p-brane. Inthis sensewecanrealizethatinthiscompactificationprocessappears a mechanism in which a higher dimensional theory that breaks translation symmetry may recover this symmetry in lower dimensions. This is in some sense in accord with the recent ideas on the possibility of breaking the translation symmetry in high energy physics. Also, the effective potentials are such that they are flatter for lower d-dimension. This issue is important in inflationary scenarios since flatter potentials may produce sufficient inflation. We can estimate the suitable d-dimensional internal space to produce sufficient inflation in our Universe (p=3)-brane. This effect may be used to probe extra dimensions in our Universe. The authors would like to thank CAPES, CNPq, and PROCAD/PNPD-CAPES for partial support. [1] D.Bazeia, J. Menezes and R. Menezes, Phys.Rev.Lett. 91 (2003) 241601, hep-th/0305234. 6 FIG. 1: Effective potential for themode ξ0 in d=3 (green) is flatter than in d=6 (blue). [2] B. Zwiebach, JHEP 0009 (2000) 028, hep-th/0008227. [3] F. A.Brito, JHEP 0508 (2005) 036, hep-th/0505043. [4] D.Bazeia, F. A. Brito and R.C. Fonseca, Eur. Phys.J. C 63 (2009) 163, 0809.3048 [hep-th]. [5] F. A.Brito and H. S.Jesuino, JHEP 1007 (2010) 031, 0912.4712 [hep-th]. [6] L. Randalland R.Sundrum,Phys.Rev.Lett. 83 (1999) 4690, hep-th/9906064. [7] R.Gregory, V. A. Rubakovand S.M. Sibiryakov,Phys.Rev.Lett. 84 (2000) 5928, hep-th/0002072. [8] G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485 (2000) 208, hep-th/0005016; G. Kofinas, E. Papantonopoulos, and I. Pappa, Phys.Rev. D 66 (2002) 104014, hep-th/0112019; G. Kofinas, JHEP 0108 (2001) 034, hep-th/0108013. [9] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D 62 (2000) 046008, hep-th/9909134; C. Csaki, J. Erlich, T.J. Hollowood and Y.Shirman,Nucl.Phys.B581 (2000) 309, hep-th/0001033;M. Cvetic,Int.J.Mod. Phys.A 16(2001) 891,hep-th/0012105;D.Z.Freedman,C.Nunez,M.SchnablandK.Skenderis,Phys.Rev.D69(2004)104027, hep-th/0312055. G. W. Gibbons and N. D. Lambert, Phys. Lett. B 488 (2000) 90, hep-th/0003197; K. Skenderis and P. K. Townsend, Phys. Lett. B 468 (1999) 46, hep-th/9909070; F. Brito, M. Cvetic and S. Yoon, Phys. Rev. D 64 (2001) 064021,hep-ph/0105010;D.Bazeia,F.A.BritoandJ.R.S.Nascimento,Phys.Rev.D68(2003)085007,hep-th/0306284; F. A.Brito, F. F. Cruz and J. F. N.Oliveira, Phys. Rev.D 71 (2005) 083516, hep-th/0502057. [10] M. Cvetic and N.D. Lambert, Phys.Lett. B 540 (2002) 301, hep-th/0205247. [11] D.Bazeia, F. A. Brito and A.R.Gomes, JHEP 0411 (2004) 070, hep-th/0411088. [12] D.Bazeia, F. A. Brito and L.Losano, JHEP 0611 (2006) 064, hep-th/0610233. [13] M. Cvetic and H. H.Soleng, Phys.Rept. 282 (1997) 159, hep-th/9604090. [14] A.Padilla, Class. Quant. Grav. 22 (2005) 681 (2005), hep-th/0406157. [15] O.Castillo-Felisola, A. Melfo, N. Pantoja and A. Ramirez, Phys. Rev.D 70 (2004) 104029, hep-th/0404083. [16] G. Gabadadze, L.Grisa and Y.Shang, JHEP 0608 (2006) 033, hep-th/0604218. [17] M. Ito, Phys.Lett. B 528 (2002) 269, hep-th/0211268.